Prior


Gbase<- function(num, mean=0, sigma=1){
return( rnorm(n=1, mean, sigma))
}
PUrn_sample<- function(n, alpha, Gbase){
#theta.1 sample from base measure
theta<- Gbase(1)
group_vec <- c(1)
for (i in 2:n){
#probability to sample from base measure
p1<- alpha/ (alpha + i-1)
#probability to sample from existing atoms
p2<- 1/ (alpha + i-1)
pvec<- c(p1, p2*group_vec)
# prob <- c(alpha/(length(count.vector)+alpha), count.vector/(length(count.vector)+M))
group_num<- sample(0:length(group_vec), prob=pvec, size=1)
if (group_num == 0){
#add new atom
theta <- c(theta, Gbase(1))
group_vec <- c(group_vec, 1)
}else{
#update existing group for group_vec
group_vec[group_num] = group_vec[group_num] + 1
}
}
return(list(theta=theta, group= group_vec))
}
alpha_0<-1
n<- 500
x<- PUrn_sample(n, alpha_0,Gbase)
k_groups<- x$group
weights<- c(k_groups/(alpha_0+n), alpha_0/(alpha_0+n))
df_weights <- data.frame(matrix(NA, nrow =length(weights), ncol =1))
df_weights$pw<-weights
df_weights$tr<-1:length(weights)
pl_weigths<- ggplot(df_weights, aes(x=tr, y=pw)) +
geom_segment( aes(x=tr,xend=tr,y=0,yend=pw)) +
geom_point( size=0.5, color="red", fill=alpha("blue", 0.3), alpha=0.4, shape=21, stroke=2)+ labs(title=paste0("Prior weights"))+
theme_bw() + theme(axis.text.x = element_text(angle = 0, hjust = 1,size = 10), strip.text = element_text(size = 15),legend.position = "top", plot.title = element_text(hjust = 0.5))
pl_weigths
df_weights <- data.frame(matrix(NA, nrow =length(weights), ncol =1))
df_weights$pw<-weights
df_weights$tr<-c(x$theta,20)
pl_weigths<- ggplot(df_weights, aes(x=tr, y=pw)) +
geom_segment( aes(x=tr,xend=tr,y=0,yend=pw)) +
geom_point( size=0.5, color="red", fill=alpha("blue", 0.3), alpha=0.4, shape=21, stroke=2)+ labs(title=paste0("Prior weights"))+
theme_bw() + theme(axis.text.x = element_text(angle = 0, hjust = 1,size = 10), strip.text = element_text(size = 15),legend.position = "top", plot.title = element_text(hjust = 0.5))
pl_weigths
PolyaUrnGenerator <- function(num, M, G0.generator) {
X.vector <- G0.generator(n = 1)
count.vector <- 1
for (i in 2:num){
prob <- c(M/(length(count.vector)+M), count.vector/(length(count.vector)+M))
category <- which(rmultinom(n=1, size=1,prob=prob)==1) - 1
if (category == 0){
X.vector <- c(X.vector, G0.generator(n = 1))
count.vector <- c(count.vector, 1)
}else{
count.vector[category] = count.vector[category] + 1
}
}
return(list(X.vector, count.vector))
}
## Define a Polya Urn sampler
PolyaUrnSampler <- function(n, M, cutoff, num = 10000, G0.generator=rnorm){
output <- matrix(NA, nrow = n, ncol = length(cutoff) + 1)
cutoff <- c(-Inf, cutoff, Inf)
for(i in 1:n){
PU.sample <- PolyaUrnGenerator(num=num, M=M, G0.generator=G0.generator)
for(j in 1:ncol(output)){
output[i,j] <- sum(PU.sample[[2]][intersect(which(PU.sample[[1]] > cutoff[j]), which(PU.sample[[1]] <= cutoff[j+1]))])/num
}
}
return(output)
}
##
x<- PolyaUrnGenerator(1,1,rnorm)
y<- PUrn_sample(1, 1, Gbase)
ks.test(f_approx$ecf, ecdf(X))
One-sample Kolmogorov-Smirnov test
data: f_approx$ecf
D = 0.79, p-value = 8.882e-16
alternative hypothesis: two-sided
Posterior_P_Urn_exact<- function(X,theta,i,alpha){
q_weights<- dnorm(X[i], theta)
r_weight<- alpha*dnorm(X[i], mean=0, sd=sqrt(2))
#sum should be one
num_norm<- sum(q_weights)+r_weight
q_weights_norm<- q_weights/num_norm
r_weight_norm<- r_weight/num_norm
label <- sample(0:length(theta),prob=c(r_weight_norm,q_weights_norm),size=1,replace=TRUE)
if (label==0){
theta_new<- rnorm(1, mean=X[i], sd=sqrt(1/2))
}
else{
theta_new <- theta[label]
}
}
sample<- gen_data(mu_vec=c(1,5),sigma_vec=rep(0.2, 2),ns=50,prob_vec=c(0.5,0.5))
X<- sample
theta.matrix <- matrix(NA, nrow = 10000, ncol = length(X))
alpha<-1
#weights.matrix <- matrix(NA, nrow = 100, ncol = length(X)+1)
theta.init<- runif(length(X),-3,3)
theta.matrix[1,] <- theta.init
for (i in 2:nrow(theta.matrix)){
theta_last<- theta.matrix[i-1,]
N <- length(X)
temp_theta <- c(theta_last, rep(NA, N))
for (j in 1:N){
temp_theta[N+j] <- Posterior_P_Urn_exact(X, temp_theta[(j+1):(j+N-1)],j,alpha)
}
theta_S<- temp_theta[(N+1):(N*2)]
theta.matrix[i,] <-theta_S
}
#burnin period
theta.final <- theta.matrix[5000:10000,]
xgrid <- seq(from= -10, to=2,length=50)
fgrid <- NULL
fbar.H <- function(xgrid,wh)
{ ## return a draw F ~ p(F | ...) (approx)
fx <- rep(0,length(xgrid))
for(h in 1:length(wh))
fx <- fx + dnorm(xgrid,m=wh[h],sd=1)
return(fx/length(wh))
}
gibbs.H <- function(n.iter=500){
xgrid <- seq(from= -10, to=10,length=50)
fgrid <- NULL
plot(density(X),xlab="X",ylab="Y",bty="l",type="l",xlim=c(-10, 10),ylim=c(0,1), main="")
for(iter in 1:nrow(theta.matrix)){
## record draw F ~ p(F | th,sig,y) (approx)
f <- fbar.H(xgrid,theta.matrix[iter,])
lines(xgrid,f,col=iter,lty=3)
fgrid <- rbind(fgrid,f)
}
## add overall average (= posterior mean) to the plot
fbar <- apply(fgrid,2,mean)
lines(xgrid,fbar,lwd=3,col=2)
}
Posterior_G<- function(n, alpha, Gbase, theta){
}
Approximation 1st order
\[\theta_{n} \mid \theta_{n-1},..\theta_1 \sim \frac{k_n \alpha}{n} H + \frac{1}{n} \sum_i^{k_n} (n_i - \alpha)\delta_{\theta^*_j}\]
Approximation 2nd order
\[\theta_{n} \mid \theta_{n-1},..\theta_1 \sim \frac{k_n \alpha + \beta_n}{n + \beta_n} H + \frac{1}{n + \beta_n} \sum_i^{k_n} (n_i - \alpha)\delta_{\theta^*_j}\]
For PY process \(\beta_n = \theta\)
polya_urn_model <- function(base_measure, N_ball, alpha) {
balls <- c()
for (i in 1:N_ball) {
if (runif(1) < alpha / (alpha + length(balls))) {
#Add a new ball color.
new_color <- base_measure()
balls <- c(balls, new_color)
} else {
#Pick out a ball from the urn, and add back a # ball of the same color
ball <- balls[sample(1:length(balls), 1)]
balls <- c(balls, ball)
} }
balls
}
alpha_vect <- c(1, 10, 100, 1000)
N_urns <- 3
sigma2 <- c(1,.4,.2)
N_draws <- 100
N_xaxis <- 200
x_axis <- seq(-3, 3, length = N_xaxis)
result <- NULL
for (alpha in alpha_vect) {
PU <- polya_urn_model(function() rnorm(1), N_draws, alpha)
for (u in 1:N_urns) {
res <- mapply(function(mean) dnorm(x_axis, rep(mean, N_xaxis),
rep(sigma2[u], N_xaxis)), PU)
res <- apply(res, 1, mean)
new_draw <- cbind(res, x_axis, alpha, sigma2[u])
result <- rbind(result, new_draw)
} }
result <- as.data.frame(result)
names(result) <- c("density", "x", "alpha", "sigma2")
DP_mixt <- qplot(data = result, y = density, x = x,
geom = c("line", "area")) +
facet_grid(alpha ~ sigma2, labeller =
label_bquote(rows = alpha == .(alpha),
cols = sigma^2 == .(sigma2))) +
aes(color = as.factor(alpha)) +
theme(legend.position = "none")
DP_mixt
Add a new chunk by clicking the Insert Chunk button on the toolbar or by pressing Cmd+Option+I.
When you save the notebook, an HTML file containing the code and output will be saved alongside it (click the Preview button or press Cmd+Shift+K to preview the HTML file).
The preview shows you a rendered HTML copy of the contents of the editor. Consequently, unlike Knit, Preview does not run any R code chunks. Instead, the output of the chunk when it was last run in the editor is displayed.
---
title: "R Notebook"
output: html_notebook
---


```{r libs, echo=FALSE}
rm(list=ls())
#Packages for gamma function estimation
library(pracma)
library(Brobdingnag)
library(copula)
library(plotly)
library(distrEx)

```

# $V_{kn}$ functions and Prior cluster distribtion.

```{r V_kn functions and Prior cluster distribtion, echo=FALSE}

############### V for PY / Dir / NGG########################
v_py<- function(kval, sigma,theta,npoints){
  c_v<-1:(kval-1)
  v_nk<- (theta +sigma*c_v)
  Vup<- prod(v_nk)
  n_vec<- 1:(npoints-1)
  Vlow<- prod(theta +n_vec)
  V_t_nk<-Vup/Vlow
  return(V_t_nk)
}

############### V for NGG########################
v_ng<- function(beta, sigma, kval, npoints){
  sum<-0
  coef_low<-as.brob(gamma(npoints))
  coef_high<-as.brob(exp(beta)* sigma^(kval-1))
  coef<- coef_high/coef_low
  incv<- as.brob(vector(length=npoints))
  for(i in (0:(npoints-1))){
    gn<- as.brob(gammainc(kval - i/sigma,beta)[2])
    #gn<- gamma_inc(kval - i/sigma,beta)
    ckn<- as.brob(choose(npoints-1,i))
    sum<- sum + ((-1)^i)*(beta^(i/sigma))*ckn*gn
    incv[i+1]<- gn
  }
  sumf<- sum/coef
  sumn<- as.numeric(sumf)
  return(list(sum=sumf, incg= incv)) 
}


v_ng2<- function(beta, sigma, kval, npoints){
  sum<-0
  incv<- as.brob(vector(length=npoints))
  for(i in (0:(npoints-1))){
    coef_low<-as.brob(gamma(npoints))
    coef_high<-as.brob(exp(beta)* sigma^(kval-1))
    coef<- coef_high/coef_low
    gn<- as.brob(gammainc(kval - i/sigma,beta)[2])
    ckn<- as.brob(choose(npoints-1,i))
    sum<- sum + ((-1)^i)*(beta^(i/sigma))*ckn*gn*coef
    incv[i+1]<- gn
  }
  sumf<- sum
  sumn<- as.numeric(sumf)
  return(list(sum=sumn, incg= incv)) 
}

###############Generalized coefficient########################

gen_fac_coef<-function(kval,sigma,npoints){
  sum<-0
  kfac<-factorial(kval)
  for(i in (0:kval)){
    n_vec<- 0:(npoints-1)
    sn<- prod(-i*sigma +n_vec)
    ckn<- choose(kval,i)
    #print((-1)^i*ckn*sn)
    sum<- sum + ((-1)^i)*ckn*sn 
  }
  sumf<- sum/kfac
  return(sumf)
}


########### density for  NGG #############################
prob_ng<- function(kg, sigma, npoints, beta){
  pb_v_all<- v_ng2(beta, sigma, kg, npoints)
  pb_v<-as.brob(pb_v_all$sum)
  pb_gen<- as.brob(gen_fac_coef(kg,sigma, npoints))
  prob<- (pb_v*pb_gen)/(as.brob(sigma^kg))
  prob_num<- as.numeric(prob)
  return(prob_num)
}

########### density for PY#############################

prob_py<- function(kg, npoints, sigma, theta){
  pb_v<- v_py(kg,sigma, theta,npoints)
  pb_gen<- gen_fac_coef(kg,sigma, npoints)
  prob<- (pb_v*pb_gen)/(sigma^kg)
  return(prob)
}

########### density for Dirichlet#############################
prob_dir<- function(k, npoints, theta){
  n_vec<- 0:(npoints-1)
  theta_n<- prod(theta +n_vec)
  prob<- ((theta^k) *(abs(Stirling1(npoints,k))))/theta_n
  return(prob)
}

prob_dir_large_dim<- function(k, npoints, theta){
  n_vec<-as.brob( 0:(npoints-1))
  theta_n<- prod(theta +n_vec)
  stir<- as.brob(abs(Stirling1(npoints,k)))
  powerk<- as.brob((theta^k))
  prob_brob<- powerk*(stir/theta_n)
  prob<- as.numeric(prob_brob)
  return(prob)
}

#####################################################################################################################################
#prib_ng(kg, npoints, sigma, beta)
k_vec<-seq(1,50,by=5)
sigma_vec<-seq(0.2,0.8, by=0.1)
z<- outer(k_vec,sigma_vec,Vectorize(prob_ng),npoints=50, beta=1)

p<- plot_ly(showscale = TRUE) %>%
  add_surface(x=k_vec, y=sigma_vec,z =z, cmin = min(z), cmax = max(z),colorbar=list(title='PY'), colorscale = list(c(0,1),c("rgb(255,112,184)","rgb(128,0,64)")),opacity = 0.98) %>%
  layout(title="Prior distribution", scene = list(xaxis= list(title="K"),yaxis= list(title="sigma"),zaxis= list(title="N",range = c(min(z),max(z)))))
p

```

# Data generation

Gaussian mixture:

$$Y \sim \frac{1}{2}N(1,0.2) +\frac{1}{2}N(5,0.2)$$

```{r data generation}
gen_data<- function(mu_vec=c(1,5,3,5), sigma_vec=rep(0.1, 4), ns=50, prob_vec=c(0.25,0.25,0.25,0.25) ){
  l<- length(mu_vec)
  components <- sample(1:l,prob=prob_vec,size=ns,replace=TRUE)
  mus <- mu_vec
  sds <- sqrt(sigma_vec)
  samples <- rnorm(n=ns,mean=mus[components],sd=sds[components])
  return(samples)
}

set.seed(123)
sample<- gen_data(mu_vec=c(1,10),sigma_vec=rep(0.2, 2),ns=10000,prob_vec=c(0.5,0.5))
hist(sample,probability=TRUE,breaks=20,col=grey(.9))


```


# Model 

\begin{equation}
\begin{aligned}
(X_i \mid (\mu_i,\tau_i)) &\stackrel{iid}{\sim} N(\mu_i,\tau_i^{-1})\\
((\mu_i, \tau_i) \mid G) &\stackrel{iid}{\sim} G \\
G &\sim \mathcal{P} .
\end{aligned}
\end{equation}



Firstly, assume $\tau_i=1$,then we have $X_i \sim N(\mu_i, 1)$, base measure $G_0 = N(0,1)$, $Y_i= (\mu_i)$. For sampling we use Polya Urn characterization

\begin{equation}
     p(Y_i | {Y_{-i}}) = \frac{\alpha}{\alpha+n-1} G_0 + \frac{1}{\alpha + n -1} \sum_{j=1}^{n-1} \delta_{Y_j} \qquad \forall i=1,\ldots ,n.
\end{equation}



Gibbs sampler:
\begin{equation}
\begin{aligned}
G &\mid \theta, X \sim DP(\alpha +n,\frac{\alpha}{(\alpha + n)} G_0 + \frac{1}{\alpha + n)}\sum_j \delta_{\theta_j)}\\
\theta_i &\mid \theta_{-i},X, G \sim \sum_{j\neq i}q_{ij} \delta_{\theta_j} + r_i G \\
q_{ij} &= b N(x_,\theta_j)\\
r_i &=  b\alpha \int f(x_i, \theta)dG_0(\theta), \text{Here: } N(x_i, \theta,1)N(\theta, 0,1)\propto N(x_i/2, 1/2).\\

\end{aligned}
\end{equation}


#Prior

```{r mixture}

Gbase<- function(num, mean=0, sigma=1){
  return( rnorm(n=1, mean, sigma))
}



PUrn_sample<-  function(n, alpha, Gbase){
  #theta.1 sample from base measure
   theta<- Gbase(1)
   group_vec <- c(1)
   for (i in 2:n){
     #probability to sample from base measure
     p1<- alpha/ (alpha + i-1)
     #probability to sample from existing atoms
     p2<- 1/ (alpha + i-1)
     pvec<- c(p1, p2*group_vec)
    # prob <- c(alpha/(length(count.vector)+alpha), count.vector/(length(count.vector)+M))
    group_num<-  sample(0:length(group_vec), prob=pvec, size=1)

    if (group_num == 0){
      #add new atom
      theta <- c(theta, Gbase(1))
      group_vec <- c(group_vec, 1)
    }else{
      #update existing group for group_vec
      group_vec[group_num] = group_vec[group_num] + 1
    }
  }
  return(list(theta=theta, group= group_vec))
}
alpha_0<-1
n<- 500
x<- PUrn_sample(n, alpha_0,Gbase)
k_groups<- x$group
weights<- c(k_groups/(alpha_0+n), alpha_0/(alpha_0+n))
  
df_weights <- data.frame(matrix(NA, nrow =length(weights), ncol =1))
df_weights$pw<-weights
df_weights$tr<-1:length(weights)
pl_weigths<- ggplot(df_weights, aes(x=tr, y=pw)) +
  geom_segment( aes(x=tr,xend=tr,y=0,yend=pw)) +
  geom_point( size=0.5, color="red", fill=alpha("blue", 0.3), alpha=0.4, shape=21, stroke=2)+  labs(title=paste0("Prior weights"))+
  theme_bw() + theme(axis.text.x = element_text(angle = 0, hjust = 1,size = 10), strip.text = element_text(size = 15),legend.position = "top", plot.title = element_text(hjust = 0.5))
pl_weigths


df_weights <- data.frame(matrix(NA, nrow =length(weights), ncol =1))
df_weights$pw<-weights
df_weights$tr<-c(x$theta,20)
pl_weigths<- ggplot(df_weights, aes(x=tr, y=pw)) +
  geom_segment( aes(x=tr,xend=tr,y=0,yend=pw)) +
  geom_point( size=0.5, color="red", fill=alpha("blue", 0.3), alpha=0.4, shape=21, stroke=2)+  labs(title=paste0("Prior weights"))+
  theme_bw() + theme(axis.text.x = element_text(angle = 0, hjust = 1,size = 10), strip.text = element_text(size = 15),legend.position = "top", plot.title = element_text(hjust = 0.5))
pl_weigths


 PolyaUrnGenerator <- function(num, M, G0.generator) {
   X.vector <- G0.generator(n = 1)
   count.vector <- 1
   for (i in 2:num){
     prob <- c(M/(length(count.vector)+M), count.vector/(length(count.vector)+M))
     category <- which(rmultinom(n=1, size=1,prob=prob)==1) - 1
     if (category == 0){
       X.vector <- c(X.vector, G0.generator(n = 1))
       count.vector <- c(count.vector, 1)
     }else{
       count.vector[category] = count.vector[category] + 1
     }
   }
   return(list(X.vector, count.vector))
 }


## Define a Polya Urn sampler
PolyaUrnSampler <- function(n, M, cutoff, num = 10000, G0.generator=rnorm){
  output <- matrix(NA, nrow = n, ncol = length(cutoff) + 1)
  cutoff <- c(-Inf, cutoff, Inf)
  for(i in 1:n){
    PU.sample <- PolyaUrnGenerator(num=num, M=M, G0.generator=G0.generator)
    for(j in 1:ncol(output)){
      output[i,j] <- sum(PU.sample[[2]][intersect(which(PU.sample[[1]] > cutoff[j]), which(PU.sample[[1]] <= cutoff[j+1]))])/num
    }
  }
  return(output)
}

## 
x<- PolyaUrnGenerator(1,1,rnorm)
y<- PUrn_sample(1, 1, Gbase)

```





```{r Gibbs theta}
Posterior_P_Urn_exact<- function(X,theta,i,alpha){
  q_weights<- dnorm(X[i], theta)
  r_weight<- alpha*dnorm(X[i], mean=0, sd=sqrt(2))
  #sum should be one
  num_norm<- sum(q_weights)+r_weight
  q_weights_norm<- q_weights/num_norm
  r_weight_norm<- r_weight/num_norm
  label <- sample(0:length(theta),prob=c(r_weight_norm,q_weights_norm),size=1,replace=TRUE)
  if (label==0){
    theta_new<- rnorm(1, mean=X[i], sd=sqrt(1/2))
  }
  else{
    theta_new <- theta[label]
  }
}


sample<- gen_data(mu_vec=c(1,10),sigma_vec=rep(0.2, 2),ns=100,prob_vec=c(0.5,0.5))
X<- sample
theta.matrix <- matrix(NA, nrow = 10000, ncol = length(X))
alpha<-1
#weights.matrix <- matrix(NA, nrow = 100, ncol = length(X)+1)
theta.init<- runif(length(X),-3,3)
theta.matrix[1,] <- theta.init
for (i in 2:nrow(theta.matrix)){
    theta_last<- theta.matrix[i-1,]
    N <- length(X)
    temp_theta <- c(theta_last, rep(NA, N)) 
    for (j in 1:N){
      temp_theta[N+j] <- Posterior_P_Urn_exact(X, temp_theta[(j+1):(j+N-1)],j,alpha)
    }
    theta_S<- temp_theta[(N+1):(N*2)]
    theta.matrix[i,] <-theta_S
}

#burnin period
theta.final <- theta.matrix[5000:10000,] 
 
#xgrid <- seq(from= -5, to=15,length=100)
#fgrid <- NULL

fbar.H <- function(xgrid,wh)
{ ## return a draw F ~ p(F | ...) (approx)
  
  fx <- rep(0,length(xgrid))
  for(h in 1:length(wh))
    fx <- fx + dnorm(xgrid,m=wh[h],sd=1)
  fx<- fx +  M/(n+M)*dnorm(xgrid,m=m0,sd=sqrt(B0+sig))
  return(fx/length(wh))
}


fbar.H_base <- function(xgrid,wh)
{ ## return a draw F ~ p(F | ...) (approx)
  fx <- rep(0,length(xgrid))
  for(h in 1:length(wh)){
    fx <- fx + dnorm(xgrid,m=wh[h],sd=1) 
  }
  fx_m<- fx/ (length(wh)+ alpha)
  fx_m<- fx_m +  alpha/(length(wh)+alpha)*dnorm(xgrid,m=0,sd=sqrt(2))
  return(fx_m)
}



ecfbar.H_base <- function(xgrid,wh)
{ ## return a draw F ~ p(F | ...) (approx)
  
  fx <- rep(0,length(xgrid))
  for(h in 1:length(wh)){
    fx <- fx + pnorm(xgrid,m=wh[h],sd=1) 
  }
  fx_m<- fx/ (length(wh)+ alpha)
  fx_m<- fx_m +  alpha/(length(wh)+alpha)*pnorm(xgrid,m=0,sd=sqrt(2))
  return(fx_m)
}


gibbs.H <- function(n.iter=10000){
  xgrid <- seq(from= -5, to=12,length=50)
  cgrid<- seq(from= -5, to=15,length=50)
  fgrid <- NULL
  ecfgrid <- NULL
  plot(density(X),xlab="X",ylab="Y",bty="l",type="l",xlim=c(-5, 15),ylim=c(0,1), main="")
  for(iter in 1:nrow(theta.final)){
      ## record draw F ~ p(F | th,sig,y) (approx)
      f   <- fbar.H_base(xgrid,theta.final[iter,])
      lines(xgrid,f,col=iter,lty=3)
      fgrid <- rbind(fgrid,f)
    }
    ## add overall average (= posterior mean) to the plot
    fbar <- apply(fgrid,2,mean)
    lines(xgrid,fbar,lwd=3,col=2)
    plot(ecdf(X),xlab="X",ylab="Y",bty="l", main="")
  for(iter in 1:nrow(theta.final)){
      ## record draw F ~ p(F | th,sig,y) (approx)
      ecf<- ecfbar.H_base(cgrid,theta.final[iter,])
      lines(cgrid,ecf,col=iter,lty=3)
      ecfgrid <- rbind(ecfgrid,ecf)
    }
    ## add overall average (= posterior mean) to the plot
    fbar <- apply(fgrid,2,mean)
    ecfbar<- apply(ecfgrid,2,mean)
    lines(cgrid,ecfbar,lwd=3,col=2)
    
    return(list(f_ap=fbar, ecf=ecfbar))
}


f_approx<- gibbs.H()
ks.test(f_approx$ecf, ecdf(X))

```










```{r Gibbs theta approx}
Posterior_P_Urn_exact<- function(X,theta,i,alpha){
  q_weights<- dnorm(X[i], theta)
  r_weight<- alpha*dnorm(X[i], mean=0, sd=sqrt(2))
  #sum should be one
  num_norm<- sum(q_weights)+r_weight
  q_weights_norm<- q_weights/num_norm
  r_weight_norm<- r_weight/num_norm
  label <- sample(0:length(theta),prob=c(r_weight_norm,q_weights_norm),size=1,replace=TRUE)
  if (label==0){
    theta_new<- rnorm(1, mean=X[i], sd=sqrt(1/2))
  }
  else{
    theta_new <- theta[label]
  }
}


sample<- gen_data(mu_vec=c(1,5),sigma_vec=rep(0.2, 2),ns=50,prob_vec=c(0.5,0.5))
X<- sample
theta.matrix <- matrix(NA, nrow = 10000, ncol = length(X))
alpha<-1
#weights.matrix <- matrix(NA, nrow = 100, ncol = length(X)+1)
theta.init<- runif(length(X),-3,3)
theta.matrix[1,] <- theta.init
for (i in 2:nrow(theta.matrix)){
    theta_last<- theta.matrix[i-1,]
    N <- length(X)
    temp_theta <- c(theta_last, rep(NA, N)) 
    for (j in 1:N){
      temp_theta[N+j] <- Posterior_P_Urn_exact(X, temp_theta[(j+1):(j+N-1)],j,alpha)
    }
    theta_S<- temp_theta[(N+1):(N*2)]
    theta.matrix[i,] <-theta_S
}

#burnin period
theta.final <- theta.matrix[5000:10000,] 
 
xgrid <- seq(from= -10, to=2,length=50)
fgrid <- NULL

fbar.H <- function(xgrid,wh)
{ ## return a draw F ~ p(F | ...) (approx)
  
  fx <- rep(0,length(xgrid))
  for(h in 1:length(wh))
    fx <- fx + dnorm(xgrid,m=wh[h],sd=1)
  return(fx/length(wh))
}


gibbs.H <- function(n.iter=500){
xgrid <- seq(from= -10, to=10,length=50)
fgrid <- NULL
plot(density(X),xlab="X",ylab="Y",bty="l",type="l",xlim=c(-10, 10),ylim=c(0,1), main="")
for(iter in 1:nrow(theta.matrix)){
    ## record draw F ~ p(F | th,sig,y) (approx)
    f   <- fbar.H(xgrid,theta.matrix[iter,])
    lines(xgrid,f,col=iter,lty=3)
    fgrid <- rbind(fgrid,f)
  }
  ## add overall average (= posterior mean) to the plot
  fbar <- apply(fgrid,2,mean)
  lines(xgrid,fbar,lwd=3,col=2)
}



```










```{r Gibbs posterior weights}

Posterior_G<- function(n, alpha, Gbase, theta){
  
}

```



Approximation 1st order

$$\theta_{n} \mid \theta_{n-1},..\theta_1 \sim \frac{k_n \alpha}{n} H + \frac{1}{n} \sum_i^{k_n} (n_i - \alpha)\delta_{\theta^*_j}$$


Approximation 2nd order

$$\theta_{n} \mid \theta_{n-1},..\theta_1 \sim \frac{k_n \alpha + \beta_n}{n + \beta_n} H + \frac{1}{n + \beta_n} \sum_i^{k_n} (n_i - \alpha)\delta_{\theta^*_j}$$

For PY process $\beta_n = \theta$







```{r}


polya_urn_model <- function(base_measure, N_ball, alpha) {
  balls <- c()
  for (i in 1:N_ball) {
    if (runif(1) < alpha / (alpha + length(balls))) { 
      #Add a new ball color.
      new_color <- base_measure()
      balls <- c(balls, new_color)
        } else {
    #Pick out a ball from the urn, and add back a # ball of the same color
          
    ball <- balls[sample(1:length(balls), 1)] 
    balls <- c(balls, ball)
    } }
    balls 
}

alpha_vect <- c(1, 10, 100, 1000)
N_urns <- 3
sigma2 <- c(1,.4,.2)
N_draws <- 100
N_xaxis <- 200
x_axis <- seq(-3, 3, length = N_xaxis)
result <- NULL
  for (alpha in alpha_vect) {
    PU <- polya_urn_model(function() rnorm(1), N_draws, alpha)
    for (u in 1:N_urns) {
    res <- mapply(function(mean) dnorm(x_axis, rep(mean, N_xaxis),
                                       rep(sigma2[u], N_xaxis)), PU)
    res <- apply(res, 1, mean)
    new_draw <- cbind(res, x_axis, alpha, sigma2[u])
    result <- rbind(result, new_draw)
} }
result <- as.data.frame(result)
names(result) <- c("density", "x", "alpha", "sigma2")
DP_mixt <- qplot(data = result, y = density, x = x,
                 geom = c("line", "area")) +
  facet_grid(alpha ~ sigma2, labeller =
               label_bquote(rows = alpha == .(alpha),
                            cols = sigma^2 == .(sigma2))) +
  aes(color = as.factor(alpha)) +
  theme(legend.position = "none")
DP_mixt
```










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